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\chapter{Results and Analyses}

Once the theoretical information have been obtained, apply all the approaches of finding stationarity region to the MiLADY campaign measured data. In the Digital Broadcasting Research Laboratory, TU Ilmenau, Access of the MiLADY campaign data is provided. 

\section{MiLADY campaign Measurement}

The S-Band measurement at 2.3 GHz data were collected from the MiLADY campaign carried out at the east coast of the USA in September 2008 \cite{MiLADY}. The measurement routes from Jacksonville (Florida) to Portland (Maine) is shown in Figure ~\ref{fig:MiLADY campaign route}.

Over the total distance of 3700 km power levels of four satellites (2 Geostationary and 2 Highly elliptical orbit satellites) with different elevation and different azimuth were measured. Antenna noise power of the signal in quiet band was measured. Two GPS receivers and two cameras were mounted in the measurement van and one fish eye camera is also mounted on roof top of the measurement van \cite{MiLADY}. 

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth,height=0.8\textwidth]{./bilder/MiLADYcampaignroute}
\end{center}
\caption{MiLADY Measurement Campaign Route \cite{MiLADY}}
\label{fig:MiLADY campaign route}
\end{figure}


For better analyzing the fast fading parameters of the channel, high resolution LMS received data (means high sampling rate). The sampling rate of received signal power level (2.1 GHz) helped us to determine fast fading parameters. Require to have more information about the propagation characteristics to develop LMS channel model. The environmental and infrastructure close to the receiver mainly effect the propagation characteristic of the land mobile satellite (LMS) channel. We have images of the whole route from the two cameras, which helped us to determine the environmental types such as: 

\begin{itemize}
\item Urban
\item Suburban
\item Rural 
\item Commercial 
\item Highway
\end{itemize}

Table ~\ref{tab:Available MiLADY measurement length for different environment} shows the amount of available measurement data from MiLADY campaign split into environmental condition and used for statistical analysis \cite{MiLADY}.


\begin{table}[htb]
\centering
\begin{tabular}{ |l | r| }
\hline
		Environment Class       & Measurement Length  \\ \hline
    Urban                   & 54 km               \\ \hline
    Suburban                & 96 km               \\ \hline
    Rural                   & 301 km              \\ \hline
    Commercial              & 59 km               \\ \hline
    Highway                 & 1227 km              \\ 
    \hline
\end{tabular}
\caption{Available MiLADY measurement length for different environment \cite{MiLADY}}
\label{tab:Available MiLADY measurement length for different environment}
\end{table}

In this section, correlation structure and stationarity region results analyzed for the MiLADY measured data as shown in Figure ~\ref{fig:Fixed distance received measurement signal} and for Artificial test data as shown in Figure ~\ref{fig:Artificial Signal Generated for test}. 



\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{./bilder/fixed_interval_measurement_linear}
\caption{Fixed distance received measurement signal}
\label{fig:Fixed distance received measurement signal}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{./bilder/artificial_linear_1000_5000}
\caption{Artificial Signal Generated for test}
\label{fig:Artificial Signal Generated for test}
\end{figure}


   
Next section discuss about to remove the fast fading effect from the LMS received signal using the sliding hamming window averaging.


\section{Fast Fading Effect Remove}

Sliding hamming window is used to remove the fast fading effect of LMS received signal envelope. Moving average filtering (zero phase filtering) is performed with window length of 15$\lambda$ for averaging over signal duration to remove fast fading effect.

The window size is chosen as recommended in \cite{filteri}. The resulting slow variations are shown by the red curve for the LMS received signal shown by blue in Figure ~\ref{fig:Slow variations for the full time-series received measurement signal} and for the LMS signal stretch results are shown in Figure ~\ref{fig:Slow variations for the fixed distance received measurement signal}.


\begin{figure}[htbp]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/Full_time_series_dB}
\caption{Slow variations for the LMS signal envelope}
\label{fig:Slow variations for the full time-series received measurement signal}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/fixed_timeseries_db_4400_4900}
\caption{Slow variations for the LMS signal stretch}
\label{fig:Slow variations for the fixed distance received measurement signal}
\end{subfigure}
\caption{Slow variations of the received measurement signal}\label{fig:Slow variations of the received measurement signal}
\end{figure}


Measured data after removal of the fast fading effect using sliding window averaging is shown in Figure ~\ref{fig:Slow variations for the full time-series received measurement signal}. In the next chapters, we analyze the results of the stationarity interval using different methods.


\section{Correlation Structure Analysis}

Correlation structure matrices for LMS measured signal from MiLADY campaign data is shown in Figure ~\ref{fig:Correlation structures analysis}.

\begin{figure}[htbp]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/fixed_interval_measurement_linear}
\caption{LMS signal stretch}
\label{Fixed distance received measurement signal}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/correlation_matrix_db_4400_4900}
\caption{Correlation Coefficient Matrix (CCM) for measured data}
\label{Correlation coefficient matrix analysis}
\end{subfigure}\\
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/cmd_dist_db_4400_4900}
\caption{Correlation Matrix Distance (CMD) for measured data}
\label{CMD analysis}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/Ncmd_dist_db_4400_4900}
\caption{Normalized Correlation Matrix Distance (NCMD) for measured data}
\label{NCMD analysis}
\end{subfigure}
\caption{Correlation structures analysis}\label{fig:Correlation structures analysis}
\end{figure}




Comparison of this correlation structure is given in the Figure ~\ref{fig:Correlation structures analysis} for the fixed signal interval as shown in Figure ~\ref{Fixed distance received measurement signal}. In this figures diagonal corresponds to correlation value between two similar points where $d_1 = d_2$ and hence for CCM it will be 1 and for CMD and NCMD it will be 0.

In all Figure ~\ref{fig:Correlation structures analysis}, we can see that in all figures the signal between 110 m to 370 m is stationary. This portion is indicated with red in Figure ~\ref{Correlation coefficient matrix analysis} and with blue in Figure ~\ref{CMD analysis},~\ref{NCMD analysis}.

As we can see in Figure ~\ref{CMD analysis},~\ref{NCMD analysis}., that there is no difference in CMD and NCMD approach for this signal because for this signal portion smallest eigenvalue is zero and because of that normalization factor $K_N \approx 1$.



\section{Effect of window length on CMD and NCMD}


For small window size M = 2, we have shown CMD and NCMD correlation matrices in Figure ~\ref{CMD analysis},~\ref{NCMD analysis}. 

In this section, we determine the performance for different window size and compare it.

Figure ~\ref{fig:Effect of window size on CMD and NCMD} shows CMD matrix and NCMD matrix with different window size (M) used to calculate the local correlation matrices $\bm{R}(k)$ and $\bm{R}(l)$.

\begin{figure}[htbp]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/CMD_5_4400_4900}
\caption{CMD with window length M = 5 for measured data}
\label{fig:CMD with window length M = 5 for measured data}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/NCMD_5_4400_4900}
\caption{NCMD with window length M = 5 for measured data}
\label{fig:NCMD with window length M = 5 for measured data}
\end{subfigure}\\
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/CMD_45_4400_4900}
\caption{CMD with window length M = 45 for measured data}
\label{fig:CMD with window length M = 45 for measured data}
\end{subfigure}~ 
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/NCMD_45_4400_4900}
\caption{NCMD with window length M = 45 for measured data}
\label{fig:NCMD with window length M = 45 for measured data}
\end{subfigure}\\
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/CMD_100_4400_4900}
\caption{CMD with window length M = 100 for measured data}
\label{fig:CMD with window length M = 100 for measured data}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/NCMD_100_4400_4900}
\caption{NCMD with window length M = 100 for measured data}
\label{fig:NCMD with window length M = 100 for measured data}
\end{subfigure}
\caption{Effect of window size on CMD and NCMD}\label{fig:Effect of window size on CMD and NCMD}
\end{figure}


For the small window size of (M = 2 or M = 5) in both CMD and NCMD approach, Stationary region is identified for some interval, but it is difficult to estimate the stationary region in the non stationary part. 

In Figure ~\ref{fig:CMD with window length M = 5 for measured data} and Figure ~\ref{fig:NCMD with window length M = 5 for measured data}, it is visible that it is still difficult to identify stationary region between 0 m to 100 m distance. 

As shown in Figure ~\ref{fig:CMD with window length M = 45 for measured data} and Figure ~\ref{fig:NCMD with window length M = 45 for measured data} for window size M = 45, shows the stationary region in CMD and NCMD is selectable between 110m to 370m and for other regions 0m to 110m non stationarity with the clear 0 correlation value or red color is indicated.

After increasing window size M = 100, it is noticeable that there is not much improvement after M = 45, but using such a big window size will increase computational complexity and may be loss of some information. 


As Figure ~\ref{fig:CMD with window length M = 100 for measured data} and Figure ~\ref{fig:NCMD with window length M = 100 for measured data} shows that using M = 100 window size also cause the reduction of stationarity interval as detected in previous window size.


From above Figure ~\ref{fig:Effect of window size on CMD and NCMD}, we can say that if window size is too small (M = 5), then non stationary region is not clearly estimated. 

For the medium window size M = 45, the stationary interval is best identified. Using the big window size (M = 100), it shows no improvement on the medium size. We can analyze our data using the medium window size for better estimation.


Window size does not make influence on the CMD and NCMD highly correlated structure, but it definitely effect the less correlated structure.


\section{Local Region of Stationarity (LRS) Analysis}



In the Figure ~\ref{fig:LRS for correlation coefficient matrix}, Usage of threshold $C_{th}$ closer to high correlated (0 for CCM and 1 for CMD,NCMD) value gives unidentified LRS. As Figure shows that $C_{th}$ = 0.9 for CCM gives unidentified LRS, but for $C_{th}$ = 0.5 gives clearly separated LRS. 

The region where correlation coefficients matrix are more than $C_{th}$(threshold) = 0.9 and for CMD and NCMD 0.1 or 0.2 is very small, so for the better and more distinguishable length of LRS we select threshold ($50\%$) value $C_{th}$ = 0.5 in Figure ~\ref{fig:LRS for Correlation Coeeficient Matrix with threshold Cth = 0.5}.



\begin{figure}[htbp]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/stationary_interval_db_09_4400_4900}
\caption{LRS for Correlation Coefficient Matrix with threshold $C_{th} = 0.9$}
\label{fig:LRS for Correlation Coefficient Matrix with threshold Cth = 0.9}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/stationary_interval_05_db_4400_4900}
\caption{LRS for Correlation Coefficient Matrix with threshold $C_{th} = 0.5$}
\label{fig:LRS for Correlation Coefficient Matrix with threshold Cth = 0.5}
\end{subfigure}
\caption{LRS for correlation coefficient matrix under the measured route distance as Figure ~\ref{fig:Fixed distance received measurement signal}}\label{fig:LRS for correlation coefficient matrix}
\end{figure}


Using this approach LRS length is defined, but accurate estimate of the start and end point of this stationarity intervals is necessary. Hence we propose a trace method, which is used to estimate a starting and ending points of this stationarity intervals.


\section{Trace-path Correlation Threshold Analysis Results}

As we discussed before that using this method, it reduces the computation time.

As mentioned in previous section that decreasing threshold value will increase the stationarity region length and also loss the accuracy.

In Figure ~\ref{fig:Stationary interval Indication for different threshold value for Measured data}, stationarity indicators plotted using different threshold values for measured data.



\begin{figure}[htbp]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/sta_indi_db_4400_4900}
\caption{Stationary Interval change indication using different threshold value without MinStateLength}
\label{fig:Stationary Interval change indication using different threshold value without Ignore value for measured data}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/sta_indi_db_4000_4900_ignor}
\caption{Stationary Interval change indication with Ignore value using different threshold value with MinStateLength}
\label{fig:Stationary interval Indication for different threshold value with Ignore value for measured data}
\end{subfigure}
\caption{Stationary interval Indication for different threshold value for Measured Data}\label{fig:Stationary interval Indication for different threshold value for Measured data}
\end{figure}



So, from this results, we can come to conclusion that it is necessary to have any selection criteria for the threshold value.


\section{Trace-path Correlation Threshold with MinStateLength Analysis and Results}

For MiLADY measured data shown in Figure ~\ref{fig:Fixed distance received measurement signal}, Stationarity change Indicator for measured signal without MinStateLength is plotted in Figure ~\ref{fig:Stationary Interval change indication using different threshold value without Ignore value for measured data} and with MinStateLength in Figure ~\ref{fig:Stationary interval Indication for different threshold value with Ignore value for measured data}.


As we can see from the Figure ~\ref{fig:Stationary Interval change indication using different threshold value without Ignore value for measured data}, that for the lower value of threshold will give some unnecessary segment indicators.

For this reason, we have implemented this approach to remove that unnecessary segment borders. This algorithm is only tested with the CCM, so at this time we are not able to give the comparison results between CCM, CMD and NCMD using this approach.

It is noticeable in Figure ~\ref{fig:Trace-path with Ignore value approach method to find starting point and End point} that 'Trace-path Correlation Threshold with MinStateLength' method has a look advance feature. MinStateLength a new parameter is added to this approach means it looks in advance up-to MinStateLength value. Thus helps to avoid single point abrupt change.

For Artificial test data shown in Figure ~\ref{fig:Artificial Signal Generated for test}, Stationary change Indicator for measured signal without MinStateLength is plotted in Figure ~\ref{fig:Stationary Interval change indication using different threshold value without Ignore value for artificial data} and with MinStateLength in Figure ~\ref{fig:Stationary interval Indication for different threshold value with Ignore value for artificial data}.


It is noticeable that Better result is achieved and also useful to remove the unwanted signal segmentation indicators. Provide better results at threshold near to high correlation, so there is no need to lower the threshold. Higher threshold means higher accuracy. In previous approach threshold up-to lower correlation is used for segmentation and still it is not able to perform as it is required.  

In Figure ~\ref{fig:Stationary interval Indication for different threshold value for Measured data}, result of stationary change indication with and without MinStateLength for Measured MiLADY data.

It is also tested for the artificial generated data for test and this approach give us very satisfying result for the test data as seen in Figure ~\ref{fig:Stationary interval Indication for different threshold value for artificial data}. 


In Figure ~\ref{fig:Stationary interval Indication for different threshold value for artificial data}, result of stationary change indication with and without MinStateLength for Artificial test data.


\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/sta_indi_db_1000_4900_arti}
\caption{Stationary Interval change indication using different threshold value without MinStateLength for Artificial Data}
\label{fig:Stationary Interval change indication using different threshold value without Ignore value for artificial data}
\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{./bilder/sta_indi_db_1000_4900_arti_ignor}
\caption{Stationary interval Indication for different threshold value with MinStateLength for artificial data}
\label{fig:Stationary interval Indication for different threshold value with Ignore value for artificial data}
\end{subfigure}
\caption{Stationary interval Indication for different threshold value for artificial data}\label{fig:Stationary interval Indication for different threshold value for artificial data}
\end{figure}



From above we can say that our proposed approach with MinStateLength works good, but still there is problem of threshold value.

After analysis of too many results we can say that it's threshold depends on the signal's standard deviation. So we can select signal's standard deviation as threshold for correlation approach. 

Still we have remained with other correlation structure to be checked for this approaches. After that We are able to compare correlation structure with this stationarity approach. 